Weak-coupling theory of high-temperature superconductivity in the antiferromagnetically correlated copper oxides

We show that the retarded interaction between quasiparticles on a two dimensional square lattice induced by the exchange of antiferromagnetic paramagnons leads uniquely to a transition to a superconducting state with ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ symmetry. We find that the effective quasiparticle interaction responsible for superconductivity possesses considerable structure in both momentum and frequency space, and show, by explicit calculations, that if one wishes to obtain quantitatively meaningful results it is essential to allow for that structure in solving the full integral equations that determine the superconducting transition temperature and the superconducting properties. With a spin-excitation spectrum and a quasiparticle-paramagnon coupling determined by fits to normal-state experiments, we obtain high transition temperatures and energy-gap behaviors comparable to those measured for ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$, ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6.63}$, and ${\mathrm{La}}_{1.85}$ ${\mathrm{Sr}}_{0.15}$ ${\mathrm{CuO}}_4$.